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On Zero Duality Gap and the Farkas Lemma for Conic Programming

Constantin Zlinescu

Faculty of Mathematics, University "Al.I.Cuza" Iai, 700506–Iai, Romania, and Institute of Mathematics, Octav Mayer, Iai, Romania
zalinesc{at}uaic.ro

Recently S. A. Clark published an interesting duality result in linear conic programming dealing with a convex cone that is not closed in which the usual (algebraic) dual problem is replaced by a topological dual with the aim of having zero duality gap under certain usual hypotheses met in mathematical finance. We present some examples to show that an extra condition is needed to reach a conclusion; this supplementary condition is also provided. We also give counterexamples for three results on hedging prices and simple proofs for two known solvability results (see Propositions 4.1 and 4.2).

Key Words: conic programming; counterexample; duality gap; Farkas lemma; hedging prices
History: Received: May 21, 2007; revision received: April 14, 2008;